Itô Diffusion

Author

John Robin Inston

Published

September 25, 2026

1 Itô Diffusions

Time-homogeneous solutions of SDEs, the processes underlying the Infinitesimal Generator, the Kolmogorov equations and the Feynman-Kac Formula.

A (time-homogeneous) Itô diffusion in \(\mathbb{R}^n\) is a stochastic process \((X_{t})\) defined on space \((\Omega, \mathcal{F},\mathbb{P})\) that satisfies the stochastic differential equation \[ \begin{cases} dX_{t}=b(X_{t})dt+\sigma(X_{t})dB_{t} \\ X_{s}=x \end{cases} \] where \(X_{t}\in \mathbb{R}^n\), \(B_{t}\in \mathbb{R}^m\), \(b:\mathbb{R}^n\to\mathbb{R}^n\), \(\sigma:\mathbb{R}^n\to \mathbb{R}^{n \times m}\) where it is assumed that \(b,\sigma\) are both Lipschitz on \(\mathbb{R}^n\) (to ensure the existence and uniqueness of strong solutions).

The distinguishing feature of diffusions is that the drift and diffusion coefficients \(b,\sigma\) are time independent and the Lipschitz condition is here to ensure the existence and uniqueness of the strong solution. Next we detail the results outlining the key properties of Itô diffusions.

Denoting the solution to the SDE above with initial condition \(X_{s}=x\) as an Itô diffusion by \(X_{s+h}^{s,x}\) then \[ \forall s \geq 0,~x \in \mathbb{R}^n,~\{ X_{s+t}^{s,x} \}_{t \geq 0}\stackrel{\mathcal{L}}{=}\{ X_{t}^{0,x} \}_{t \geq 0}. \]

Proof (sketch): The increments \(B_{s+t}-B_{s}\), \(t\geq 0\) form a BM, so \(X^{s,x}_{s+\cdot}\) solves the same SDE as \(X^{0,x}_{\cdot}\) driven by a BM. Since \(b,\sigma\) do not depend on time, uniqueness of strong solutions gives equality in law.

For an Itô diffusion \(X\) we have \[ \forall t,h \geq 0,~ X_{h}^{0,X_{t}^{0,x}}\stackrel{a.s.}{=}X_{t+h}^{0,x}. \]

Proof: By construction, both are strong solutions to the same SDE \[ X_{t+h}^{0,x}=X_{t}^{0,x}+\int_{t}^{t+h}{b(X_{s}^{0,x})}~d{s}+\int_{t}^{t+h}{\sigma(X_{s}^{0,x})}~d{B_{s}}. \] By uniqueness of strong solutions the two processes are indistinguishable.

2 Further Properties

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